The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Clustering COMP Research Seminar BCB 713 Module Spring 2011 Wei Wang

COMP Data Mining: Concepts, Algorithms, and Applications 2 Drawback of Distance-based Methods Hard to find clusters with irregular shapes Hard to specify the number of clusters Heuristic: a cluster must be dense

COMP Data Mining: Concepts, Algorithms, and Applications 3 Directly Density Reachable Parameters Eps: Maximum radius of the neighborhood MinPts: Minimum number of points in an Eps- neighborhood of that point NEps(p): {q | dist(p,q)  Eps} Core object p: |Neps(p)|  MinPts Point q directly density-reachable from p iff q  Neps(p) and p is a core object p q MinPts = 3 Eps = 1 cm

COMP Data Mining: Concepts, Algorithms, and Applications 4 Density-Based Clustering: Background (II) Density-reachable Directly density reachable p 1  p 2, p 2  p 3, …, p n-1  p n  p n density-reachable from p 1 Density-connected Points p, q are density-reachable from o  p and q are density-connected pq o p q p1p1

COMP Data Mining: Concepts, Algorithms, and Applications 5 DBSCAN A cluster: a maximal set of density- connected points Discover clusters of arbitrary shape in spatial databases with noise Core Border Outlier Eps = 1cm MinPts = 5

COMP Data Mining: Concepts, Algorithms, and Applications 6 DBSCAN: the Algorithm Arbitrary select a point p Retrieve all points density-reachable from p wrt Eps and MinPts If p is a core point, a cluster is formed If p is a border point, no points are density- reachable from p and DBSCAN visits the next point of the database Continue the process until all of the points have been processed

COMP Data Mining: Concepts, Algorithms, and Applications 7 Problems of DBSCAN Different clusters may have very different densities Clusters may be in hierarchies

COMP Data Mining: Concepts, Algorithms, and Applications 8 OPTICS: A Cluster-ordering Method OPTICS: ordering points to identify the clustering structure “Group” points by density connectivity Hierarchies of clusters Visualize clusters and the hierarchy

COMP Data Mining: Concepts, Algorithms, and Applications 9 DENCLUE: Using Density Functions DENsity-based CLUstEring Major features Solid mathematical foundation Good for data sets with large amounts of noise Allow a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significantly faster than existing algorithms (faster than DBSCAN by a factor of up to 45) But need a large number of parameters

COMP Data Mining: Concepts, Algorithms, and Applications 10 Grid-based Clustering Methods Ideas Using multi-resolution grid data structures Use dense grid cells to form clusters Several interesting methods STING WaveCluster CLIQUE

COMP Data Mining: Concepts, Algorithms, and Applications 11 STING: A Statistical Information Grid Approach The spatial area area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution

COMP Data Mining: Concepts, Algorithms, and Applications 12 STING: A Statistical Information Grid Approach (2) Each cell at a high level is partitioned into a number of smaller cells in the next lower level Statistical information of each cell is calculated and stored beforehand and is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell count, mean, s, min, max type of distribution—normal, uniform, etc. Use a top-down approach to answer spatial data queries Start from a pre-selected layer — typically with a small number of cells For each cell in the current level compute the confidence interval

COMP Data Mining: Concepts, Algorithms, and Applications 13 STING: A Statistical Information Grid Approach (3) Remove the irrelevant cells from further consideration When finish examining the current layer, proceed to the next lower level Repeat this process until the bottom layer is reached

COMP Data Mining: Concepts, Algorithms, and Applications 14 STING: A Statistical Information Grid Approach (4) Advantages: Query-independent, easy to parallelize, incremental update O(K), where K is the number of grid cells at the lowest level Disadvantages: All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected

COMP Data Mining: Concepts, Algorithms, and Applications 15 WaveCluster A multi-resolution clustering approach which applies wavelet transform to the feature space A wavelet transform is a signal processing technique that decomposes a signal into different frequency sub- band. Both grid-based and density-based Input parameters: # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform.

COMP Data Mining: Concepts, Algorithms, and Applications 16 WaveCluster How to apply wavelet transform to find clusters Summaries the data by imposing a multidimensional grid structure onto data space These multidimensional spatial data objects are represented in an n-dimensional feature space Apply wavelet transform on feature space to find the dense regions in the feature space Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse

COMP Data Mining: Concepts, Algorithms, and Applications 17 Wavelet Transform Decomposes a signal into different frequency subbands. (can be applied to n- dimensional signals) Data are transformed to preserve relative distance between objects at different levels of resolution. Allows natural clusters to become more distinguishable

COMP Data Mining: Concepts, Algorithms, and Applications 18 What Is Wavelet (2)?

COMP Data Mining: Concepts, Algorithms, and Applications 19 Quantization

COMP Data Mining: Concepts, Algorithms, and Applications 20 Transformation

COMP Data Mining: Concepts, Algorithms, and Applications 21 WaveCluster Why is wavelet transformation useful for clustering Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary

COMP Data Mining: Concepts, Algorithms, and Applications 22 WaveCluster Effective removal of outliers

COMP Data Mining: Concepts, Algorithms, and Applications 23 WaveCluster Multi-resolution Cost efficiency

COMP Data Mining: Concepts, Algorithms, and Applications 24 WaveCluster

COMP Data Mining: Concepts, Algorithms, and Applications 25 WaveCluster Major features: Complexity O(N) Detect arbitrary shaped clusters at different scales Not sensitive to noise, not sensitive to input order Only applicable to low dimensional data

COMP Data Mining: Concepts, Algorithms, and Applications 26 CLIQUE (Clustering In QUEst) Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space CLIQUE can be considered as both density-based and grid-based It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into non-overlapping rectangular units A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter A cluster is a maximal set of connected dense units within a subspace

COMP Data Mining: Concepts, Algorithms, and Applications 27 CLIQUE: The Major Steps Partition the data space and find the number of points that lie inside each cell of the partition. Identify the subspaces that contain clusters using the Apriori principle Identify clusters: Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests. Generate minimal description for the clusters Determine maximal regions that cover a cluster of connected dense units for each cluster Determination of minimal cover for each cluster

COMP Data Mining: Concepts, Algorithms, and Applications 28 CLIQUE Salary (10,000) age age Vacation (week)

COMP Data Mining: Concepts, Algorithms, and Applications 29 age Vacation Salary 3050  = 3 CLIQUE

COMP Data Mining: Concepts, Algorithms, and Applications 30 Strength and Weakness of CLIQUE Strength It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces It is insensitive to the order of records in input and does not presume some canonical data distribution It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness The accuracy of the clustering result may be degraded at the expense of simplicity of the method

COMP Data Mining: Concepts, Algorithms, and Applications 31 Constrained Clustering Constraints exist in data space or in user queries Example: ATM allocation with bridges and highways People can cross a highway by a bridge

COMP Data Mining: Concepts, Algorithms, and Applications 32 Clustering With Obstacle Objects Taking obstacles into accountNot Taking obstacles into account

COMP Data Mining: Concepts, Algorithms, and Applications 33 Outlier Analysis “One person’s noise is another person’s signal” Outliers: the objects considerably dissimilar from the remainder of the data Examples: credit card fraud, Michael Jordon, etc Applications: credit card fraud detection, telecom fraud detection, customer segmentation, medical analysis, etc

COMP Data Mining: Concepts, Algorithms, and Applications 34 Statistical Outlier Analysis Discordancy/outlier tests 100+ tests proposed Data distribution Distribution parameters The number of outliers The types of expected outliers Example: upper or lower outliers in an ordered sample

COMP Data Mining: Concepts, Algorithms, and Applications 35 Drawbacks of Statistical Approaches Most tests are univariate Unsuitable for multidimensional datasets All are distribution-based Unknown distributions in many applications

COMP Data Mining: Concepts, Algorithms, and Applications 36 Depth-based Methods Organize data objects in layers with various depths The shallow layers are more likely to contain outliers Example: Peeling, Depth contours Complexity O(N  k/2  ) for k-d datasets Unacceptable for k>2

COMP Data Mining: Concepts, Algorithms, and Applications 37 Distance-based Outliers A DB(p, D)-outlier is an object O in a dataset T s.t. at least fraction p of the objects in T lies at a distance greater than distance D from O Algorithms for mining distance-based outliers The index-based algorithm, the nested-loop algorithm, the cell-based algorithm

COMP Data Mining: Concepts, Algorithms, and Applications 38 Index-based Algorithms Find DB(p, D) outliers in T with n objects Find an objects having at most  n(1-p)  neighbors with radius D Algorithm Build a standard multidimensional index Search every object O with radius D If there are at least  n(1-p)  neighbors, O is not an outlier Else, output O

COMP Data Mining: Concepts, Algorithms, and Applications 39 Pros and Cons of Index-based Algorithms Complexity of search O(kN 2 ) More scalable with dimensionality than depth- based approaches Building a right index is very costly Index building cost renders the index-based algorithms non-competitive

COMP Data Mining: Concepts, Algorithms, and Applications 40 A Naïve Nested-loop Algorithm For j=1 to n do Set count j =0; For k=1 to n do if (dist(j,k)<D) then count j ++; If count j <=  n(1-p)  then output j as an outlier; No explicit index construction O(N 2 ) Many database scans

COMP Data Mining: Concepts, Algorithms, and Applications 41 Optimizations of Nested-loop Algorithm Once an object has at least  n(1-p)  neighbors with radius D, no need to count further Use the data in main memory as much as possible Reduce the number of database scans

COMP Data Mining: Concepts, Algorithms, and Applications 42 References (1) R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98 M. R. Anderberg. Cluster Analysis for Applications. Academic Press, M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure, SIGMOD’99. P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scientific, 1996 M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases. KDD'96. M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing techniques for efficient class identification. SSD'95. D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2: , D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems. In Proc. VLDB’98. S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98. A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.

COMP Data Mining: Concepts, Algorithms, and Applications 43 References (2) L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98. G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, P. Michaud. Clustering techniques. Future Generation Computer systems, 13, R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94. E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc Int. Conf. on Pattern Recognition, G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering approach for very large spatial databases. VLDB’98. W. Wang, J. Yang, R. Muntz, STING: A Statistical Information Grid Approach to Spatial Data Mining, VLDB’97. T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very large databases. SIGMOD'96.